Andres del Junco

A simple map with no prime factors

An ergodic measure-preserving transformationT of a probability space is said to be simple (of order 2) if every ergodic joining λ ofT with itself is eitherμ×μ or an off-diagonal measureμS, i.e.,μS(A×B)=μ(A∩S;−n;B) for some invertible, measure preservingS commuting withT. Veech proved that ifT is simple thenT is a group extension of any of its non-trivial factors. Here we construct an example of a weakly mixing simpleT which has no prime factors. This is achieved by constructing an action of the countable Abelian group ℤ⊕G, whereG=⊕i=1∞ ℤ2, such that the ℤ-subaction is simple and has centralizer coinciding with the full ℤ⊕G-action.

A family of counterexamples in ergodic theory

LetTα be the translationx↦x+α (mod 1) of [0, 1), α irrational. LetT be the Lebesgue measure-preserving automorphism ofX=[0, 3/2) defined byTx = x + 1 forx∈[0, 1/2),Tx=Tα(x−1) forx∈[1,3/2) andTx = Tαx forx∈[1/2, 1), i.e.T isTα with a tower of height one built over [0, 1/2). If α is poorly approximable by rationals (there does not exist {pn/qn} with |α−pn/qn|=o(qn−2)) and λ is a measure onXk all of whose one-dimensional marginals are Lebesgue and which is ⊗i − 1kT1 invariant and ergodic (l>0) then λ is a product of off-diagonal measures. This property suffices for many purposes of counterexample construction. A connection is established with the POD (proximal orbit dense) condition in topological dynamics.

Disjointness of Measure-Preserving Transformations, Minimal Self-Joinings and Category

In the coarse topology on the group of measure-preserving transformations of a Lebesgue probability space, the class of transformations disjoint from a given ergodic transformation is a dense Gδ. The class of transformations T such that the family {Ti: i ϵ ℤ} is disjoint is also a dense Gδ. As a corollary there exists an uncountable family {Tα: α ϵ A} of weakly-mixing transformations such that the family \( \{ {\text{T}}_{\text{a}}^{\text{i}}:\alpha \in {\text{A,i}} \in {\Bbb Z} - \{ 0\} \} \) is disjoint.

On minimal self-joinings in topological dynamics

If X is a compact metric space and T a homeomorphism of X we say ( X , T ) has almost minimal power joinings (AMPJ) if there is a dense G δ X * in X such that for each finite set k , x ∈( X *) k and l: k → ℤ−{0}, the orbit closure cl { } is a product of off-diagonals (POOD) on X k . By an offdiagonal on X k ′, k′ k we mean a set of the form (⊗, j∈k′ T m(j) )Δ, Δ the diagonal in X k′ , m:k′→ℤ any function, and by a POOD on X k we mean that k is split into subsets k′, on each X k′ we put an off-diagonal and then we take the product of these. We show that examples of AMPJ exist and that this definition leads to a theory completely analogous to Rudolphs theory of minimal self-joinings in ergodic theory. In particular if ( X, T ) has AMPJ the automorphism group of T is { T n }, T has only almost 1-1 factors (other than the trivial one) and the automorphism group and factors of ⊕ i ∊ k T, k finite or countably infinite, can be very explicitly described. We also discuss ℝ-actions.

Multiple Mixing and Rank One Group Actions

Suppose G is a countable, Abelian group with an element of infinite order and let X be a mixing rank one action of G on a probability space. Suppose further that the Folner sequence {Fn} indexing the towers of X satisfies a “bounded intersection property”: there is a constant p such that each {Fn} can intersect no more than p disjoint translates of {Fn}. Then X is mixing of all orders. When G = Z, this extends the results of Kalikow and Ryzhikov to a large class of “funny” rank one transformations. We follow Ryzhikov’s joining technique in our proof: the main theorem follows from showing that any pairwise independent joining of k copies of X is necessarily product measure. This method generalizes Ryzhikov’s technique.

Finitary coding of Markov random fields

SummaryThere exists a finitary code from any stationary ergodic Markov random field to any i.i.d. random field of strictly lower entropy.

Moving averages of ergodic processes

AbstractA necessary and sufficient condition for the almost everywhere convergence of the “moving” ergodic averages

Finitary codes between one-sided Bernoulli shifts

(\Phi (n))^{ - 1} \mathop \Sigma \limits_{i = n - \Phi (n) + 1}^n x_E (T^i x)